The Stacks project

Lemma 106.11.4. Let $f : \mathcal{X} \to \mathcal{Y}$ and $h : \mathcal{U} \to \mathcal{X}$ be morphisms of algebraic stacks. Assume that $\mathcal{Y}$ is locally Noetherian, that $f$ is locally of finite type and quasi-separated, that $h$ is of finite type, and that the image of $|h| : |\mathcal{U}| \to |\mathcal{X}|$ is dense in $|\mathcal{X}|$. If given any $2$-commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-u \ar[d]_ j & \mathcal{U} \ar[r]_ h & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^-y \ar@{..>}[rru] & & \mathcal{Y} } \]

where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to f \circ h \circ u$, the category of dotted arrows is either empty or a setoid with exactly one isomorphism class, then $f$ is separated.

Proof. We have to prove $\Delta $ is a proper morphism. Assume first that $\Delta $ is separated. Then we may apply Lemma 106.11.3 to the morphisms $\mathcal{U} \to \mathcal{X}$ and $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. Observe that $\Delta $ is quasi-compact as $f$ is quasi-separated. Of course $\Delta $ is locally of finite type (true for any diagonal morphism, see Morphisms of Stacks, Lemma 101.3.3). Finally, suppose given a $2$-commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-u \ar[d]_ j & \mathcal{U} \ar[r]_ h & \mathcal{X} \ar[d]^\Delta \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^-y \ar@{..>}[rru] & & \mathcal{X} \times _\mathcal {Y} \mathcal{X} } \]

where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to \Delta \circ h \circ u$. By Morphisms of Stacks, Lemma 101.41.1 and the assumption in the lemma we find there exists a unique dotted arrow. This proves the last assumption of Lemma 106.11.3 holds and the result follows.

In the general case, it suffices to prove $\Delta $ is separated since then we'll be back in the previous case. In fact, we claim that the assumptions of the lemma hold for

\[ \mathcal{U} \to \mathcal{X} \quad \text{and}\quad \Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X} \]

Namely, since $\Delta $ is representable by algebraic spaces, the category of dotted arrows for a diagram as in the previous paragraph is a setoid (see for example Morphisms of Stacks, Lemma 101.39.2). The argument in the preceding paragraph shows these categories are either empty or have one isomorphism class. Thus $\Delta $ is separated. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E95. Beware of the difference between the letter 'O' and the digit '0'.